Any physical quantity, for instance length, can appear as an argument of a nonlinear function that can be developed in a Taylor series. So your example would be identical for any other quantity not only for angle. I can make an analog computing element where a voltage is equal to the sinus of another voltage, so after your theory, voltage is dimensionless.
The reason why this is possible is that the arguments of such nonlinear functions are either explicitly or implicitly not the physical quantities, but their numeric values, i.e. the ratios between those quantities and their units, which are dimensionless.
In the case of the nonlinear sinus function, what is usually written as sin(x) is just one member of a family of functions where the arguments are angles implicitly divided by units of plane angle:
sin(x) is the sinus function with the angle implicitly divided by 1 radian
sin(x * Pi/2) is the sinus function with the angle implicitly divided by 1 right angle
sin(x * Pi*2) is the sinus function with the angle implicitly divided by 1 cycle a.k.a. turn
sin(x * Pi/180) is the sinus function with the angle implicitly divided by 1 sexagesimal degree
It is very sad that the logical thinking about angles of most people has been perverted by what they have been taught in school, which is just a bunch of nonsense copied again and again from one textbook to another.
This to me sounds like the most natural explanation. For example, in a sibling comment someone mentioned that "you can calculate e^(-t)", but I disagree: in physics it's always e^(-t / T), where T is some time constant, so that the argument of the exponential is dimensionless. Same applies to sin(x): usually we write something like sin(2pi f t), where the units of f and t cancel out, and the 2pi is there to cancel out the invisible implicit 1 radian. sin(ft) would be wrong, at t = 1 / f you wouldn't have advanced by a full cycle.
360 - (360^3)/6 = -7M degrees
2*pi - (2 * pi)^3 / 6 = -35 radians = -2k degrees
1 - (1^3)/6 = 0.8 turns = 300 degrees
As far as you three examples go, which is "correct" depends on what you are trying to calculate - if you want this to approximate the power series for sin close to 0 you should use radians. Otherwise you use something else.
A dimensional formula of a quantity just writes its unit as a function of the fundamental units.
In any equality of physical quantities, in the two sides not only the dimensionless numeric values must be equal, but also the units must be equal, which is usually expressed by saying that the dimensions must be the same, and it is verified by writing in both sides the dimensional formulae, i.e. the units of both sides as functions of the fundamental units.
A dimensionless quantity is a ratio of two quantities that are measured by the same unit, so that the units simplify during the division.
There may be different but related dimensionless quantities, which are differentiated by different definitions of those quantities, but a dimensionless quantity cannot have different units.
This is just meaningless mumbo-jumbo that has been sadly introduced in the documents of the International System of Units, in 1995, after a shameful vote of the delegates, who have voted automatically, without thinking or discussing, a vote equivalent with establishing by vote that 2 + 2 = 5.
As another example for the second assertion, you can compute e(-t) via power series too, adding seconds to seconds squared and seconds cubed, etc, which comes up all the time. But that doesn't mean `dimensionless + seconds + seconds^2` implies seconds are dimensionless any more than sin's series with `angle + angle^3` implies that angles are dimensionless.
sin(x)
= x/(1 radian)! - x³/(3 radians)! + …
= x/(1 radian) - x³/(1 radian × 2 radians × 3 radians) + …